| 摘要 |
One embodiment of the present invention provides a system for finding the roots of a system of nonlinear equations within an interval vector X=(X<SUB>1</SUB>, . . . , X<SUB>n</SUB>), wherein the system of non-linear equations is specified by a vector function f=(f<SUB>1</SUB>, . . . , f<SUB>n</SUB>). The system operates by receiving a representation of the interval vector X (which is also called a box), wherein for each dimension, i, the representation of X<SUB>i </SUB>includes a first floating-point number, alpha<SUB>i</SUB>, representing the left endpoint of X<SUB>i</SUB>, and a second floating-point number, b<SUB>i</SUB>, representing the right endpoint of X<SUB>i</SUB>. Next, the system performs an interval Newton step on X to produce a resulting interval vector, X', wherein the point of expansion of the interval Newton step is a point, x, within the interval X, and wherein performing the interval Newton step involves evaluating f(x) to produce an interval result f<SUP>1</SUP>(x). The system then evaluates a first termination condition, wherein the first termination condition is TRUE if: zero is contained within f<SUP>1</SUP>(x), J(x,X) is regular (wherein J(x,X) is the Jacobian of the function fevaluated with respect to x over the box X); and X is contained within X'. If the first termination condition is TRUE, the system terminates the interval Newton method and records X' as a final bound.
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